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The pie rule merely prevents the pie-slicer from cheating; it won't
magically make him any better at game design.

If you have two players who both understand the game very well and your
problem is that neither of them trusts the other to create a fair initial
position, this will solve your problem.

But if a designer already intends to make a balanced game and is having
trouble with the execution of that intent (which is the assumption I see in
most discussions about how to balance games), this doesn't help you at
all.

Why I disagree here is this: The pie rule drives someone to make the pie
slices as even as possible, which is what you want to do when trying to
balance.  It isn't a solution to guarantee there will be balance, BUT it
does make for a game within a game, that makes trying more interesting.

Actually, that's very often not what it does. The pie rule only works when both parties are highly adept at their assigned tasks. If the slicer is poor at balancing but the chooser is good at gauging each side's strength, then the slicer cannot help but include some imbalance that the chooser will exploit, and thus places himself at a disadvantage no matter what he does. Conversely, if the slicer is good at design but the chooser is poor at evaluation, then the slicer may deliberately provide misleading cues and trick the chooser into selecting the weaker side.

Thus, in many cases, applying the pie rule to game-balancing encourages confusing and misleading design decisions, to minimize the odds that the other party can correctly identify or utilize the advantages of each side. I think that would generally be regarded as undesirable.

But even in the best possible case, it only motivates you to balance the game, it doesn't provide any tools for doing so.

"The pie rule only works when both parties are highly adept at their
assigned tasks." 

But adeptness at their assigned tasks is simply the ability to evaluate the
quality of a board position as being likely to favor one side or another,
and that's the essence of playing skillfully: choosing moves that create
positions where you have the advantage. It isn't unfair if the pie rule
leaves an advantage with the better player. The better player naturally has
an advantage at every point in the game.

Is it unfair if it amplifies the inherent advantage one player already has? More to the point, is it desirable for it to amplify advantages in this way?

Also, I think there are real differences between the skills required to play well, to evaluate positions accurately, and to design an interesting and balanced opening position. There's certainly some overlap, but it's also quite possible to be noticeably better or worse at one of those things than the others. I could perhaps make a better guess at the material value of many fairy pieces than some chess grandmasters could; that wouldn't imply I could beat them with those pieces.

On a related note, if there is a particular piece that I'm very good at using and that my opponent is not so good at using, I can give myself an advantage by giving that pieces to BOTH sides. If I were playing against a chess grandmaster, I certainly wouldn't give both sides FIDE pieces. In fact, I would be tempted to give one side FIDE pieces, and then deliberately make the other side stronger, in the hope that my opponent will just choose the familiar pieces that he knows how to use.

The whole pie-cutting problem just becomes a whole lot messier once the question of skill comes up.

If a game is faced with uncertainty in knowing what side has an advantage,
and the only way to evaluate is based on skill of the players in the game,
why not turn this aspect into a game element itself?  Why is there a
problem with having a game more clearly differentiate between the two
players in the game, in a way that is measured by skill, as opposed to
luck?

Nothing is wrong with adding any gameplay element, skill-based or
otherwise, if the players feel it enhances the game.

But the subject of this thread does not advertise a skill-based gameplay
element, it advertises a method for achieving game balance.  Those are
completely different things!  If I'm acting as a game designer, I don't want a technique I use for balancing my game to change the nature of that game or alter the skills required to win.

The balancing I am referring to is offsetting weaknesses intrinsic in a
side in a game, NOT the skill level of the players.  The approach is to
give the players an abilities to adjust the game configuration so it is
more fair, in a way that requires a player to factor things in.  Because
the approach requires multiple rounds to be played, both players have a
chance to try to balance things.

That is the idea here, not as some sort of handicapping.

I never thought it was supposed to be a handicapping system, I thought it
was supposed to be a system for creating an initial board position from
which a typical player would have equal chances of winning regardless of
which side he played.

So I am pointing out that:

1)  At best, this sytem merely provides MOTIVATION for creating such a
position.  That's kind of like adverising a "method for lifting heavy
objects" and then revealing that the method is to offer a higher wage to
employees who can lift more.  You aren't solving the problem, you're
hiring someone else to solve it for you; the actual problem still needs to
be solved by someone at some point.

2)  Although this method aligns the player's incentives with the goal in
an idealized case, there are many realistic cases in which the player can
actually gain a larger advantage by strategically UNbalancing the game,
rather than making it as balanced as possible.  So you need to carefully
consider your circumstances before employing it, even just as a motivator.

What I believe is an important thing to keep in mind, when looking at
solutions, is to realize that solutions don't always fit every context. 
What I wrote about here doesn't guarantee to fit everything, but I do
believe it can fit a number of places.

In regards to what is discussed here, I hope what is described can be
recognized at least as a skill players can develop and test.  This skill is
the ability to evaluate and create situations that work to their advantage.
 The pie rule approach ends up providing a means to have this happen in, in
a way that would be in control of players.

Ok, let's apply the pie rule idea I have, modified, towards your game, For
the Crown.  As it is now, players would randomize what cards they have in
the game.  How about we consider the following:  One player gives a list of
X number of cards they don't want to see used.  The other player then
comes up with the cards they want the game configured, and the other player
then picks what side they want to be.

Idea here is you end up thinning down things and try to reach a state that
is agreeable to both players to play, even if it would give one player an
advantage. 

As for said to hand the problem off to other people to solve, I believe a
solution is superior if it is fluid and adjust, rather than a fixed
solution.  In short, if the solution is a framework able to handle what
players do, it is superior.

Again, to sum up, there are often multiple solutions for play balancing. 
Here I list a number of them:
http://boardgamegeek.com/blogpost/9808/are-balanced-sides-have-same-chance-of-winning-rea

* Have players bid for sides, using the bidding as a mechanism to give the
weaker sides and advantage, or handicap the stronger side. This bidding is
a game unto itself of sorts, and challenges players to evaluate sides.
* Have players play all sides and evaluate the differences in how the sides
were played. 
* Play the game teams and use a duplicate format. In games with luck, like
cardgames like Bridge on a tournament level, to account for the inherit
unfairness of the luck, they use a duplicate format where the same
conditions pass around and players play them all and they are evaluated.
Unlike the last idea, where individuals play all sides, this is done team
style.
* Fixed handicapping of one side, by giving it less resources or time to
play, or one side to count a draw as a minor victory.
* Use the pie rule. In this option, one player configures the game
conditions and the other player(s) then pick what side they want to play.
This also provides a game of sorts, where you try to figure out what would
be fairest in set up, and then decide what is the best options.